3.1.45 \(\int (b \tan ^4(c+d x))^n \, dx\) [45]

Optimal. Leaf size=59 \[ \frac {\, _2F_1\left (1,\frac {1}{2} (1+4 n);\frac {1}{2} (3+4 n);-\tan ^2(c+d x)\right ) \tan (c+d x) \left (b \tan ^4(c+d x)\right )^n}{d (1+4 n)} \]

[Out]

hypergeom([1, 1/2+2*n],[3/2+2*n],-tan(d*x+c)^2)*tan(d*x+c)*(tan(d*x+c)^4*b)^n/d/(1+4*n)

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Rubi [A]
time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3739, 3557, 371} \begin {gather*} \frac {\tan (c+d x) \left (b \tan ^4(c+d x)\right )^n \, _2F_1\left (1,\frac {1}{2} (4 n+1);\frac {1}{2} (4 n+3);-\tan ^2(c+d x)\right )}{d (4 n+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*Tan[c + d*x]^4)^n,x]

[Out]

(Hypergeometric2F1[1, (1 + 4*n)/2, (3 + 4*n)/2, -Tan[c + d*x]^2]*Tan[c + d*x]*(b*Tan[c + d*x]^4)^n)/(d*(1 + 4*
n))

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 3557

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[x^n/(b^2 + x^2), x], x, b*Tan[c + d
*x]], x] /; FreeQ[{b, c, d, n}, x] &&  !IntegerQ[n]

Rule 3739

Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Di
st[(b*ff^n)^IntPart[p]*((b*Tan[e + f*x]^n)^FracPart[p]/(Tan[e + f*x]/ff)^(n*FracPart[p])), Int[ActivateTrig[u]
*(Tan[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

Rubi steps

\begin {align*} \int \left (b \tan ^4(c+d x)\right )^n \, dx &=\left (\tan ^{-4 n}(c+d x) \left (b \tan ^4(c+d x)\right )^n\right ) \int \tan ^{4 n}(c+d x) \, dx\\ &=\frac {\left (\tan ^{-4 n}(c+d x) \left (b \tan ^4(c+d x)\right )^n\right ) \text {Subst}\left (\int \frac {x^{4 n}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\, _2F_1\left (1,\frac {1}{2} (1+4 n);\frac {1}{2} (3+4 n);-\tan ^2(c+d x)\right ) \tan (c+d x) \left (b \tan ^4(c+d x)\right )^n}{d (1+4 n)}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 53, normalized size = 0.90 \begin {gather*} \frac {\, _2F_1\left (1,\frac {1}{2}+2 n;\frac {3}{2}+2 n;-\tan ^2(c+d x)\right ) \tan (c+d x) \left (b \tan ^4(c+d x)\right )^n}{d+4 d n} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*Tan[c + d*x]^4)^n,x]

[Out]

(Hypergeometric2F1[1, 1/2 + 2*n, 3/2 + 2*n, -Tan[c + d*x]^2]*Tan[c + d*x]*(b*Tan[c + d*x]^4)^n)/(d + 4*d*n)

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Maple [F]
time = 0.20, size = 0, normalized size = 0.00 \[\int \left (b \left (\tan ^{4}\left (d x +c \right )\right )\right )^{n}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*tan(d*x+c)^4)^n,x)

[Out]

int((b*tan(d*x+c)^4)^n,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((tan(d*x+c)^4*b)^n,x, algorithm="maxima")

[Out]

integrate((b*tan(d*x + c)^4)^n, x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((tan(d*x+c)^4*b)^n,x, algorithm="fricas")

[Out]

integral((b*tan(d*x + c)^4)^n, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (b \tan ^{4}{\left (c + d x \right )}\right )^{n}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((tan(d*x+c)**4*b)**n,x)

[Out]

Integral((b*tan(c + d*x)**4)**n, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((tan(d*x+c)^4*b)^n,x, algorithm="giac")

[Out]

integrate((b*tan(d*x + c)^4)^n, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\left (b\,{\mathrm {tan}\left (c+d\,x\right )}^4\right )}^n \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*tan(c + d*x)^4)^n,x)

[Out]

int((b*tan(c + d*x)^4)^n, x)

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